A Surface Phase Field Model for Two-Phase Biological Membranes

Abstract. We study vesicles formed by lipid bilayers that are governed by an elastic bending energy and on which the lipids laterally separate forming two different phases. The energy laden phase interfaces may be modeled as curves on the hyper-surface representing the membrane (sharp interface model). The phase field methodology is another powerful tool to model such phase separation phenomena where thin layers describe the interfaces (diffuse interface model). For both approaches we characterize equilibrium shapes in terms of the Euler-Lagrange equations of the total membrane energy subject to constraints on the area of the two phases and the volume. We further show by matching appropriate formal asymptotic expansions that the sharp interface model is obtained from the diffuse interface model as the thickness of the phase interface tends to zero. The essential challenge lies in the fact that also the geometry of the membrane is unknown and depends on a small parameter representing the interface thickness.